Type-2 Fuzzy Sets and
Systems
Outline
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Introduction
Type-2 fuzzy sets.
Interval type-2 fuzzy sets
Type-2 fuzzy systems.
History
What is a T2 FS and How is
it Different From a T1 FS?
• T1 FS: crisp grades of membership
• T2 FS: fuzzy grades of membership, a fuzzy-fuzzy set.
Type-2 fuzzy sets
• Blur the boundaries of a
T1 FS
• Possibility assigned –
could be non-uniform
• Clean things up
• Choose uniform
possibilities – interval
type-2 FS
Where Does a T2 FS Come From?
• Consider a FS as a model for a word
• Words mean different things to different
people.
• So, we need a FS model that can capture
the uncertainties of a word.
• A T2 FS can do this.
• Let’s see how.
Collect Data from
a Group of Subjects
• “ On a scale of 0–10 locate the end points
of an interval for some eye contact”
Collect Data from
a Group of Subjects
“ On a scale of 0–10 locate the end points of
an interval for some eye contact”
Create a Multitude of T1 FSs
• Choose the shape of the MF, as we do for T1
FSs, e.g. symmetric triangles
• Create lots of such triangles that let us cover the
two intervals of uncertainty
Fill-er-in and Some New Terms
• UMF: Upper membership function (MF)
• LMF: Lower MF
• Shaded region: Footprint of uncertainty (FOU)
Weighting the FOU
• Non-uniform secondary MF: General T2 FS
• Uniform secondary MF: Interval T2 FS
More Terms
Type 2 fuzzy sets
• Imagine blurring a type 1 membership
function.
• There is no longer a single value for the
membership function for any x value, there
are a few
Quite tall
e.g. Tallness
e.g. Joe Bloggs
We are only interested
Type-n Fuzzy Sets
In type 2 for now
• A fuzzy set is of type n, n = 2, 3, . . . if its
membership function ranges over fuzzy sets
of type n-1. The membership function of a
fuzzy set of type-1 ranges over the interval
[0,1].
Zadeh, L.A., The Concept of a Linguistic
Variable and its Application to Approximate
Reasoning - I, Information Sciences, 8,199–
249, 1975
Type 2 fuzzy sets
• These values need not all be the same
• We can therefore assign an amplitude
distribution to all of the points
• Doing this creates a 3-D membership function
i.e. a type 2 membership function
• This characterises a fuzzy set
Example of a type 2 membership function.
The shaded area is called the ‘Footprint of
Uncertainty’ (FOU)
μA~(x,u)
jx is the set of possible u values, i.e.
j3 = [0.6, 0.8]
Jx is called the primary membership
of x and is the domain of the
secondary membership function.
The amplitude of the ‘sticks’ is
called a secondary grade
μA~(x,u)
Referring to the diagram, the
secondary membership function at
x = 1 is a /0 + b /0.2 + c /0.4.
Its primary membership values at
x = 1 are u = 0, 0.2, 0.4, and their
associated secondary grades are
a, b and c respectively.
(Mendel, 2001, p85)
FOU continued
• The FOU is the union of all primary
memberships
• It is the region bounded by all of the ‘j’ values
i.e. the red shaded region on the earlier slide.
• FOU is useful because:
– Focuses our attention on uncertainties
(blurriness!)
– Allows us to depict a type 2 fuzzy set graphically
in 2 dimensions instead of 3.
– The shaded FOUs imply the 3rd dimension on top
of it.
Type-2 Fuzzy Sets - Notation
x,u intersection
somewhere in the FOU
For all u contained in
our primary memberships
/domain
Type-2 Fuzzy Sets - Notation
Important Representations
of an IT2 FS: 1
• Vertical Slice Representation—Very
useful and widely used for computation
Important Representations
of an IT2 FS: 2
• Wavy Slice Representation—Very useful and
widely used for theoretical developments
Example
Calculating the number of
embedded sets
In the above example there would be:
5 x 5 x 2 x 5 x 5 = 1250
So there are 1250 embedded sets
More formally:
Fundamental Decomposition Theorem
N is discretisation of x
M is discretisation of u
i.e. the union of all
the embedded sets
Indicates how to
calculate the number
of embedded sets
example on previous slide
Important Representations
of an IT2 FS
• Wavy Slice: Also known as “Mendel- John
Representation Theorem (RT)”
– Importance: All operations involving IT2 FSs
can be obtained using T1 FS mathematics
• Interpretation of the two
representations: Both are covering
theorems, i.e., they cover the FOU
Set-Theoretic Operations
Centroid of type-2 fuzzy sets
Comparison with type-1
• Type-1 fuzzy sets are two dimensional
• Type-2 fuzzy sets are three dimensional
• Type one membership grades are in [0,1]
• We can have linguistic grades with type-2
• Type-1 fuzzy systems are computationally
cheap
• Type-2 fuzzy systems
are computationally
Various approaches
being taken
expensive (but..) To solve/get round this
Interval T2 FSs
• Rest of tutorial focuses exclusively on IT2 FSs
– Computations using general T2 FSs are very costly
– Many computations using IT2 FSs involve only
interval arithmetic
– All details of how to use IT2 FSs in a fuzzy logic
system have been worked out
– Software available
– Lots of applications have already occurred
Other FOUs
Interval Valued type-2 fuzzy
sets
When the amplitudes of of the
secondary membership function
all equal 1, we have an interval
valued fuzzy set.
Interval valued type-2 fuzzy
sets
A~( x, u)

~
A
i.e.
( x, u) = 1
Interval valued type-2 fuzzy
sets
A lot of researchers
use IVFS as a way
of resolving
computational
expense issue.
IVFS in 2-dimensions
Type-2 person FS
Type-2 person FS
Type-2 fuzzy system overview
So this is extra
Compared to type-1
Fuzzification
Fuzzifying in type-1
(fairly easy)
Fuzzifying in type-2
(not so easy)
Interval Type-2 FLS
• Rules don’t change, only the antecedent and consequent FS models
change
• Novel Output Processing: Going from a T2 fuzzy output set to a
crisp output—type-reduction + defuzzification
Interpretation for an IT2 FLS
• A T2 FLS is a collection of T1 FLSs
IT2 FLS Inference for One Rule
IT2 FLS Inference to Output for
Two Fired Rules
Output Processing
• Defuzzification is trivial once typereduction has been performed